Here is the text of my 20 minute talk at the ASL panel "does mathematics
need new axioms" at the ASL meeting in Urbana.
DOES MATHEMATICS NEED NEW AXIOMS?
ASL Meeting, Urbana
Panel discussion: June 5, 2000
Harvey M. Friedman
The point of view of the set theory community is well represented here. I
want to concentrate on the perspective of mathematicians outside set theory.
1. MATHEMATICIANS' VIEWPOINT.
New axioms are needed in order to settle various mathematically natural
questions. Yet no well known mathematician outside set theory is even
considering adopting any new axioms for mathematics, even though they are
aware of at least the existence of the independence results.
The difference in perspective of set theorists versus mathe-maticians who
are not set theorists, is enormous. Recall that mathematics goes back, say,
2,500 years - whereas set theory in the relevant sense dates back to the
turn of the 20th century.
For 2,500 years, mathematicians have been concerned with matters of
counting and geometry and physical notions.
These main themes gave rise to arithmetic, algebra, geometry, and analysis.
The interest in and value of mathematics is judged by mathematicians in
terms of its relevance and impact on the main themes of mathematics.
It is generally recognized by most mathematicians that set theory is the
most convenient vehicle for achieving rigor in mathematics.
For this purpose, there has evolved a more or less standard set theoretic
interpretation of mathematics, with ZFC generally accepted as the current
gold standard for rigor.
It is simply false that a number theorist is interested in and respects set
theory just as they are interested in and respect group theory, topology,
differential geometry, real and complex analysis, operators on Hilbert
space, etcetera.
The reason for this attitude is quite fundamental and extremely important.
A number theorist is of course interested in complex analysis because he
uses it so much. But not so with operators on Hilbert space. Yet there is
still a distant respect for this because of a web of substantive and varied
interconnections that chain back to number theory. Set theory does not have
comparable interconnections.
For the skeptical, the degree of extreme isolation can be subjected to
various tests including citation references - broken down even into their
nature and quality. Using a critical notion from statistics, set theory is
an extreme outlier.
Nor is set theory regarded as intrinsically interesting to mathematicians,
independent of its lack of impressive inter-connections. Why?
For the mathematician, set theory is regarded as a convenient way to
provide an interpretation of mathematics that supports rigor. A natural
number is obviously not a set, an ordered pair is obviously not a set, a
function is obviously not a set of ordered pairs, and a real number is
obviously not a set of rationals.
For the mathematician, mathematics is emphatically not a branch of set
theory. The clean interpretation of mathematics into set theory does not
commit the mathematician to viewing problems in set theory as problems in
mathematics.
The mathematician therefore evaluates set theory in terms of how well it
serves its purpose - providing a clean, simple, coherent, workable way to
formalize mathematics.
This point of view hardened as many mathematicians experimented for several
decades with what has come to be known as set theoretic problems which
turned out to be independent of ZFC.
There was a growing realiza-tion that the cause of these difficulties was
excessive generality in the formulations of the problems which allowed for
pathological cases which were radically different in character than normal
mathema-tical examples. That if the problems were formulated in more
concrete ways that still covered all known interesting cases, then the
difficulties completely disappeared.
Furthermore, distinctions between these set theoretic problems causing
difficulties and the most celebrated theorems and open problems in
mathematics can be given FORMALLY. This is in terms of quantifier
complexity and the closely related matter of absoluteness. Thus set theory
comes out as an extreme outlier which can be documented FORMALLY.
2. THE MALIGNED AXIOM OF CONSTRUCTIBILITY - MORE IS LESS AND LESS IS MORE.
The set theorist is looking for deep set theoretic phenomena, and so V = L
is anathema since it restricts the set theoretic universe so drastically
that all sorts of phenomena are demonstrably not present. Furthermore, for
the set theorist, any advantage that V = L has in terms of power can be
obtained with more powerful axioms of the same rough type that accomodate
measurable cardinals and the like - e.g., V = L(mu), or the universe is an
inner model of a large cardinal.
However, for the normal mathematician, since set theory is merely a vehicle
for interpreting mathematics so as to establish rigor, and not
mathematically interesting in its own right,
the less set theoretic difficulties and phenomena the better.
I.e., less is more and more is less. So if the mathematician were concerned
with the set theoretic independence results - and they generally are not -
then V = L is by far the most attractive solution for them.
This is because it appears to solve all set theoretic problems (except for
those asserting the existence of sets of unrestricted cardinality), and is
also demonstrably relatively consistent.
Set theorists also say that V = L has implausible consequences - e.g.,
there is a Sigma-1-2 well ordering of the reals, or there are nonmeasurable
PCA sets.
The set theorists claim to have a direct intuition which allows them to
view these as so implausible that this provides "evidence" against V = L.
However, mathematicians disclaim such direct intuition about complicated
sets of reals. Many say they have no direct intuition about all
multivariate functions from N into N!
3. QUESTION ANSWERED BY CLASSICAL DESCRIPTIVE SET THEORY?
The classical descriptive set theory coming from large car-dinals is most
often cited by set theorists as the reason why mathematics needs large
cardinal axioms. I have several objections to this claim.
a. Part of the argument is that large cardinals are needed to establish
these results. But large cardinals are not needed to establish an
alternative series of such results. E.g., V = L provides another, entirely
different, set of answers to these questions. The set theorists answer
saying V = L gives the wrong answers and large cardinals give the right
answers, citing their direct intuition about projective sets of reals. I am
very dubious about this direct intuition. I don't have it, and
mathematicians in general disclaim it.
b. Another part of the argument is that, in light of a,
set theory needs large cardinals
and therefore
mathematics needs large cardinals.
But this inference depends on a reading of our question that makes this
tautological.
Reading the question this way simply avoids the really interesting
questions, replacing them by much less interesting questions. For instance,
it avoids questions of how and under what circumstances the general
mathematical community or individual mathematicians will adopt new axioms,
should adopt new axioms, and if so, how this will be manifested.
Here is the closest I can come to the set theorists' point of view on our
question.
There is an interesting notion of "general set theory in its maximal
conceivable form" and that V = L has no basis in this context. However, the
notion is at present virtually completely unexplained, and no work that I
have seen provides any serious insight into what this really means. We
simply do not know how to explicate any relevant notion of maximality.
I agree that
"general set theory in its maximal conceivable form" needs large cardinals
axioms
is very likely to be true. But I can't conclude even that
set theory needs large cardinal axioms
let alone
mathematics needs large cardinal axioms.
4. GENERAL PREDICTIONS.
The picture is going to change radically with the new Boolean relation
theory and related developments, joining the issue of new axioms and the
relevance of large cardinals in a totally new and unexpect-edly convincing
way.
Because of the thematic nature of these developments, and the interaction
with nearly all areas of mathematics, large cardinal axioms will begin to
be accepted as new axioms for mathematics - with controversy. Use of them
will still be noted, at least in passing, for quite some time, before full
acceptance.
5. CIRCUMSTANCES SURROUNDING ACTUAL ADOPTION OF NEW AXIOMS.
The cirumstances that I envision are a coherent body of consequences of
large cardinals of a new kind.
a. They should be entirely mathematically natural. This standard is very
high for a logician trying to uncover such consequences, yet is routinely
met in mathematics (set theory included) by professionals at all levels of
achievment.
b. They should be concrete. At least within infinitary discrete
mathematics. Most ideally, involving polynomials with integer coefficients,
or even finite functions on finite sets of integers.
c. They should be thematic. If they are isolated, they will surely be
stamped as curiosities, and the math community will find a way to attack
them through an ad hoc raising of the standards for being entirely natural.
However, if they are truly thematic, then the theme itself must be
attacked, which may be difficult to do. For instance, the same theme may
already be inherent in well known basic, familiar, and useful facts.
d. They should have points of contact with a great variety of mathematics.
e. They should be open ended. I.e., the pain will never end until the
adoption of large cardinals.
f. They should be elementary. E.g., at the level of early undergraduate or
gifted high school. That way, even scientists and engineers can relate to
it, so it is harder for the math community to simply bury them.
g. Their derivations should be accessible, with identifiable general
techniques. This way, the math community can readily immerse itself in
hands on crystal clear uses of large cardinals that beg to be removed - but
cannot.
We have omitted an additional curcumstance:
h. They should be used in normal mathematics as pursued be-fore such
thematic results.
For some mathematicians, h will be required before they consider the issue
really joined. I already know that for some well known core
math-ematicians, h is definitely not required - that the issue is already
sufficiently
joined for them by Boolean relation theory.
Implicit in criteria a-g is that the body of examples and the theme launch
a new area, with an eventual AMS classifi-cation number. This new area will
be accepted as part of the general unremovable furniture of contemporary
mathematics whose intrinsic interest is comparable to other established
areas in mathematics. In this way, the issue of large cardinal axioms will
be joined for a critical number of important core mathematicians.
6. LARGE CARDINALS OR THEIR 1-CONSISTENCY?
The statements coming out of Boolean relation theory are
provably equivalent to the 1-consistency of large cardinals. So instead of
adopting the large cardinal axioms themselves, one can instead adopt their
1-consistency.
When put into proper perspec-tive, this is more of a criticism of form than
over substance. Adopting large cardinals amounts to asserting
"every consequence of large cardinals is true."
Adopting the 1-consistency of large cardinals amounts to asserting
"every Pi-0-2 consequence of large cardinals is true."
The obviously more natural choice is to accept large cardinals, since the
latter is syntactic and not an attractive axiom candidate.
However, for the purposes of proving Pi-0-2 sentences, these two choices
are essentially equivalent.
Another consideration is more practical. When the working mathematician
wants to develop Boolean relation theory, the proofs are incomparably more
direct and mathematically elegant when done under the assumption of the
large cardinal axioms themselves than under the 1-consistency.
When I publish that "phi needs large cardinals to prove" I explicitly
formalized this as "any reasonable formal system that proves phi must
interpret large cardinals in the sense of Tarski." This gives a precise
sense to "needs."
There is an interesting point of some relevance here. Statements in Boolean
relation theory are also consequences of the existence of a real valued
measurable cardinal - a related kind of large cardinal axiom.
Let me put it somewhat differently. There is a substantial and coherent
list of non syntactic axiom candidates, in-cluding large cardinal axioms
and other axioms. In this list, only certain axiom candidates settle
questions in Boolean relation theory. The most appropriate ones from
various points of view are in fact the small large cardinal axioms. That
is the obvious move to make from the point of view of a working scientist.
If they later prove to be inconsistent, then we can undergo theory
revision. The key advance is that the issue of new axioms finally promises
to get joined in a serious way for the mathematics community.
APPENDIX
Two open questions in set theory.
The following are relevant to the panel discussion.
a. Prove that large cardinals provides a complete theory of the projective
hierarchy.
b. Prove that there are no "simple" axioms that settle the continuum
hypothesis.